Question
Upstudy AI Solution
Tutor-Verified Answer
Answer
A solução da equação \( 5^{x-1} = \frac{1}{625} \) é \( x = -3 \).
Solution
1. Primeiramente, escrevemos \( \frac{1}{625} \) em função de uma potência de 5. Sabemos que \( 625 = 5^4 \), logo:
\[
\frac{1}{625} = \frac{1}{5^4} = 5^{-4}
\]
2. Assim, a equação fica:
\[
5^{x-1} = 5^{-4}
\]
3. Com as bases iguais, podemos igualar os expoentes:
\[
x-1 = -4
\]
4. Resolvendo para \( x \):
\[
x = -4 + 1 = -3
\]
Portanto, a solução da equação é \( x = -3 \).
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Beyond the Answer
To solve the equation \( 5^{x-1} = \frac{1}{625} \), we begin by recognizing that \( 625 \) can be rewritten as a power of \( 5 \). Specifically, \( 625 = 5^4 \), which allows us to represent \( \frac{1}{625} \) as \( 5^{-4} \). Thus, we can rewrite the equation as \( 5^{x-1} = 5^{-4} \). Since the bases are the same, we can set the exponents equal to each other: \[ x - 1 = -4 \] Now, solving for \( x \) gives: \[ x = -4 + 1 = -3 \] The solution of the equation is \( x = -3 \). In summary, the final answer is: \[ \boxed{-3} \]
